On locally finite ordered rooted trees and their rooted subtrees

We first revisit and generalize a known result about a doubly exponential sequence that describes the number of $k$-ary ordered rooted trees of height $h$ where $k\ge 2$ is a fixed integer. Such a sequence has the form ${\left(t_k\left(h\right)\right)}_{h\ge 0}$ where $t_k\left(h\right)=\left(c{\left(k\right)}^{k^h}\right)-1$ for each given $k$ and $c\left(k\right)\in$ $R$. We provide the first detailed analysis of the real number sequence ${\left(c\left(k\right)\right)}_{k\ge 2}$ and show, in particular, that this sequence is strictly decreasing and has a limit 2 when $k$ tends to infinity. We then turn our attention to a more general setting for sequences that describe the number of ordered rooted trees of a given height. This has applications in algorithm analyses where searches in rooted trees are performed. We consider infinite ordered rooted trees in which each ordered rooted subtree induced by all the vertices on levels $h\in$ $N$or less is a finite ordered rooted tree of height $h$ of a certain type. In particular, we study those infinite trees $T$ in which each vertex has infinitely many descendants. We first give a complete characterization of those infinite trees for which the number $s\left(T_h\right)$ of ordered rooted subtrees $T_n$ of height at most $h$ of $T$ is bounded by a polynomial in $h$. We then present natural lower and upper bounds for $s\left(T_h\right)$ and use those to obtain a tight threshold function $f\left(h\right)$ for which we have $s\left(T_h\right)=\Theta \left(h^{f\left(h\right)}\right)$ for infinitely many $h\in N$. This threshold function can be presented in terms of the Lambert W function; the upper branch of the two inverses of the function $w\mapsto w{e}^w$. Finally, we investigate some theoretical properties of $s\left(T_h\right)$ for those infinite trees $T$ that have a finite width when viewed as partially ordered sets (posets) with the root as the sole maximum element. In particular, we show that for large enough $h$ the function $s\left(T_h\right)$ is given by a polynomial in $h$ and we determine the degree and the leading coefficient of this polynomial.